Nelly Villamizar — Page 2

arXiv2014

Bounds on the dimension of trivariate spline spaces: A homological approach

We consider the vector space of globally differentiable piecewise polynomial functions defined on a three-dimensional polyhedral domain partitioned into tetrahedra. We prove new lower and upper bounds on the dimension of this space by applying homological techniques. We give an insight into different ways of approaching this problem by exploring its connections with the Hilbert series of ideals generated by powers of linear forms, fat points, the so-called Fröberg-Iarrobino conjecture, and the weak Lefschetz property.

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arXiv2012

Homological techniques for the analysis of the dimension of triangular spline spaces

The spline space $C_k^r(Δ)$ attached to a subdivided domain $Δ$ of $\R^{d} $ is the vector space of functions of class $C^{r}$ which are polynomials of degree $\le k$ on each piece of this subdivision. Classical splines on planar rectangular grids play an important role in Computer Aided Geometric Design, and spline spaces over arbitrary subdivisions of planar domains are now considered for isogeometric analysis applications. We address the problem of determining the dimension of the space of bivariate splines $C_k^r(Δ)$ for a triangulated region $Δ$ in the plane. Using the homological introduced by Billera (1988), we number the vertices and establish a formula for an upper bound on the dimension. There is no restriction on the ordering and we obtain more accurate approximations to the dimension than previous methods and furthermore, in certain cases even an exact value can be found. The construction makes also possible to get a short proof for the dimension formula when $k\ge 4r+1$, and the same method we use in this proof yields the dimension straightaway for many other cases.

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